In an apparatus of this type, known from U.S. Pat. No. 4,782,684, Benton, the bearer for the bending members has a driven rotatable disc which is mounted on the frame so as to be rotatable in both directions about the axis of feed and is provided with a passage aperture for the rod-shaped material. Extending from the rotatable disc is an arm, which is provided with a slide guide which is penetrated by the material and is aligned with the rotatable disc aperture. At its free end, the bearer bears a bending head which is fitted with bending mandrels which can be rotated or swivelled about an axis perpendicular to the guide. The known design primarily has the disadvantage that bars with relatively large limb lengths cannot be effectively supported by the bearer arm and are at risk of becoming deformed under their own weight. On the other hand, due to the lack of adequate support, no possibility is provided of counteracting or compensating for a spring-back tendency of the material during bending. Finally, in the known construction, a cutting-off device for the bent element is provided on a separate frame away from the bending head, which cutting-off device is complicated in construction and susceptible to malfunction.